A nonlinear dynamics for risk contagion: analyzing the not risk-free equilibrium

In this paper we carry out the stability analysis of the not risk-free steady state which is involved in a financial contagion dynamics. Starting from an analogy between economic sectors and ecosystems, the Susceptible-Infected-Recovered (SIR) approach is employed to describe the risk dynamics by a nonlinear differential system with time delay. A main assumption is that contagion phenomenon is modelled by a Holling Type II functional response so that an incubation time for risk infection is accounted for; moreover, after contagion, some agents may be recovered from high risk and get a temporary immunity for a temporal period represented by the time delay characterizing the dynamics. We perform the analysis around the not risk-free equilibrium in terms of asymptotic stability and point out the crucial role of the incubation time and the financial immunity period in establishing whether risk crisis continues to exist in the economic sector at the long run or it can be eliminated.